Parity check codes can be considered as codes having low density and parity checking. The person skilled in the art is aware of examples of such codes under the name LDPC (Low Density Parity Check) codes.
LDPC codes were introduced by Gallager in 1962 and rediscovered in 1996 by MacKay and Niel. These codes play a fundamental role in modern communications, and in particular, because of their very good error correcting performance.
The LDPC codes generally used are binary LDPC codes. That is, they are defined on a Galois field of order 2. The word field is understood to mean in the mathematical sense, and it is recalled that a Galois field is a field containing a finite number of elements. A binary LDPC code defined on a Galois field of order 2 comprises symbols capable of taking just two values, for example, the values 0 or 1.
From a theoretical viewpoint, non-binary LDPC codes are additionally known. They are defined on a Galois field of order strictly greater than 2, for example, equal to an integer power of 2 strictly greater than 1.
These non-binary LDPC codes are beneficial since their performance in terms of error correction can be increased significantly with respect to binary LDPC codes. These binary LDPC codes are more robust, and in particular, with strings of symbols to be encoded of small sizes.
The performance gain obtained with non-binary LDPC codes is accompanied by a significant increase in the decoding complexity as well as the memory size necessary for the decoder. Specifically, for example, for an LDPC code defined in a Galois field of order 2p, it is necessary to store messages of size 2p and, at the present time, this problem has not been adequately addressed.